To determine whether a triangle can exist with side lengths of 1 cm, 13 cm, and a third side, we should use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Given sides:
- Side 1 (a) = 1 cm
- Side 2 (b) = 13 cm
- Side 3 (c) = x (unknown)
We will analyze this based on the possible values of x.
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The sum of the lengths of the two shorter sides must be greater than the length of the longer side:
- \( 1 + x > 13 \) (This implies \( x > 12 \))
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The sum of the lengths of the two sides, taking the longest side to be x:
- \( 1 + 13 > x \) (This implies \( x < 14 \))
From these inequalities, we find:
- \( 12 < x < 14 \)
The only whole number that satisfies this condition is \( x = 13 \).
Now, this means the sides would be:
- Side 1 = 1 cm
- Side 2 = 13 cm
- Side 3 = 13 cm
Since we have two sides that are equal (13 cm each), this triangle would be classified as isosceles.
Thus, the answer is isosceles.